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Optimal inventory control with state-dependent jumps

Part of: Stochastic analysis Stochastic systems and control General theory in ordinary differential equations

Published online by Cambridge University Press:  24 March 2025

Lijun Bo  and
Yijie Huang
Lijun Bo*
Affiliation:
Xidian University
Yijie Huang*
Affiliation:
University of Science and Technology of China
*
*Postal address: School of Mathematics and Statistics, Xidian University, Xi’an, 710126, China. Email: [email protected]
**Postal address: School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China. Email: [email protected]
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Abstract

We study an optimal inventory control problem under a reflected jump–diffusion netflow process with state-dependent jumps, in which the intensity of the jump process can depend on the inventory level. We examine the well-posedness of the associated integro-differential Hamilton–Jacobi–Bellman (ID-HJB) equation with Neumann boundary condition in the classical sense. To achieve this, we first establish the existence of viscosity solutions to the ID-HJB equation of an auxiliary control problem with a compact policy space, which is proved to be equivalent to the primal problem. We reformulate the ID-HJB equation as a Neumann HJB equation with the (non-local) integral term expressed in terms of the value function of the auxiliary problem and prove the existence of a unique classical solution to the Neumann HJB equation. Then, the well-posedness of the primal ID-HJB equation follows from the unique classical solution of the Neumann HJB equation and the existence of viscosity solutions to the auxiliary ID-HJB equation. Based on this classical solution, we characterize the optimal (admissible) inventory control strategy and show the verification result for the primal control problem.

Keywords

Inventory controlstate-dependent jumpsintegro-differential Hamilton–Jacobi–Bellman equationNeumann boundaryviscosity solutionclassical solution

MSC classification

Primary: 93E20: Optimal stochastic control 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 93E03: Stochastic systems, general 34A34: Nonlinear equations and systems, general
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 32
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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